We've already found for the kinetic moment:
L² Yl,m =
h² l(l +1) Yl,m
and LzYl,m =
h m Yl,m
where L² is the squared kinetic moment
operator and h²l(l +1) is the eigenvalue
of eigenfunction Yl,m.
The z-component has its own eigenvalue which is equal to
hm.
Now let's have a look at spin operators and eigenvalues. Here we can write down the following:
s² χ = h²
s(s + 1) χ
and szχ = h
msχ
where h²s(s + 1) is the eigenvalue of the squared
spin and hms is the eigenvalue of its
z-component. We obtain for s = ½ the following eigenvalues ms =
± ½ with corresponding eigenfunctions χ+ (for ms = +½) and
c- (for ms = −½):
s² χ± = ¾
h²χ±
and szχ± = ± ½
h χ±
The Dirac style for spin description is very visual:
χ+ = ( | 1 | ) = |+> | c- = ( | 0 | ) = |-> | |
0 | 1 |
sx =
|
0 1 | ) |
1 0 | ||
sy =
i |
0 −i | ) |
i 0 | ||
sz =
|
1 0 | ) |
0 −1 |
One can easily write down for |+> and |-> <+| = (1
0) <-| =
(0 1) for instance
|+> <-| = ( | 1 0 | ) .( | 0 1 | ) = ( | 0 1 | ) |
0 0 | 0 0 | 0 0 |
Certainly we must know the matrix multiplication !
s² = sx² + sy² + sz² = ¾
|
1 0 | ) |
0 1 |
s² |+> = ¾
|
1 0 | ) ( | 1 | ) = ¾ h² ( | 1 | ) = ¾
|
0 1 | 0 | 0 | ||||
s² |-> = ¾
|
1 0 | ) ( | 0 | ) = ¾ h² ( | 0 | ) = ¾
|
0 1 | 1 | 1 | ||||
sz |+>
= |
1 0 | ) ( | 1 | ) =
|
1 | ) =
|
0 −1 | 0 | 0 | ||||
sz |->
= |
1 0 | ) ( | 0 | ) =
|
0 | ) =
− |
0 −1 | 1 | −1 |
Sometimes the designation χms is used for ms = +½ and ms = −½.
The total wavefunction Y in the radial symmetrical field is then as follows:
Ynlmlms= Rnl(r) Ylml(J,j) cms
and an electron can be described by four quantum numbers n, l, ml und ms.
The total kinetic moment is the sum
of orbital moment
and intrinsic kinetic moment (Spin)
of an electron:
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Auf diesem Webangebot gilt die Datenschutzerklärung der TU Braunschweig mit Ausnahme der Abschnitte VI, VII und VIII.